The Logic of Belief and Belief Change: A Decision...

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Journal of Economic Theory � ET2131

journal of economic theory 69, 1�23 (1996)

The Logic of Belief and Belief Change:A Decision Theoretic Approach

Stephen Morris*

Department of Economics, University of Pennsylvania,3718 Locust Walk, Philadelphia, Pennsylvania 19104

Savage [L. Savage, ``The Foundations of Statistics,'' Wiley, New York, 1954]showed how properties of a decision maker's probabilistic beliefs can be deducedfrom primitive consistency axioms on preferences. This paper extends that approachand shows how logical properties of belief��which underlie economists' models ofinformation and knowledge��can be related to properties of preferences. Journal ofEconomic Literature Classification Number: D80. � 1996 Academic Press, Inc.

1. Introduction

Savage [25] showed how properties of a decision maker's probabilisticbeliefs can be deduced from primitive consistency axioms on preferences.1

However the laws of probability theory are not the only properties ofbelief that are relevant for economists. More basic logical properties ofbelief underlie the concepts of ``information'' and ``knowledge'' used byeconomists. The use of partitions to represent information entails the``positive introspection'' property that if something is believed, it is believedthat it is believed, as well as other much stronger logical properties dis-cussed below. It is a weakness of economic theory that standard ways ofmodeling information are never related to underlying decision theoreticaxioms.

The purpose of this paper is to take the logic of the Savage approach onestep further, and deduce such logical properties of beliefs, and rules forchanging them, from preferences. It is possible to deduce properties of how

article no. 0035

10022-0531�96 �18.00

Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* An earlier version of this paper was circulated under the title ``Revising Knowledge:A Decision Theoretic Approach,'' as CARESS Working Paper 92-27. This version has beenconsiderably improced by valuable and detailed comments on earlier versions from MichaelBacharach, Philippe Mongin, Hyun Song Shin, an anonymous referee, and an associate editorof this journal. The current version substantially reflects those comments, although I alone amresponsible for any remaining errors. I gratefully acknowledge financial support for thiswork from a University of Pennsylvania Research Foundation Grant and National ScienceFoundation Grant *SES-9308515.

1 Savage's approach followed Ramsey [22] and de Finetti [8].


decision makers' beliefs vary across states of the world and through time byconsidering axioms on how decision makers' preferences (and thus theirchoices) vary across states and through time.

A decision maker is said to believe an event E if he is indifferent betweenevery action which yields the same outcome whenever E is true. Thus beliefis defined as a property of preferences. A decision maker is assumed to havea preference ordering over acts with uncertain consequences at every stateof the world, and at every date. Thus at state |, I may care about whathappens if state |$ occurs. But at state |$, I may have some differentpreferences over acts. The purpose of this paper is to consider, in thisframework, the relation between properties of belief and properties ofpreferences.

This approach yields an immediate reward in the first result of thispaper. The assumption that an individual's preference relation is always acomplete ordering is shown to be equivalent to the assumption that theindividual's system of beliefs is normal. Normality is the most basic logicalproperty of a system of beliefs in the possible world semantics of Kripke[14] and Hintikka [13]. Suppose that belief is generated by anaccessibility, or possibility, relation specifying which states are believedpossible in which other states. An event E is believed at state | if E con-tains every state thought possible at state |. A system of beliefs is said tobe normal if it can be derived from such a possibility relation.

In logical treatments of belief and knowledge, further properties of beliefsand knowledge (beyond normality) are added. In order to derive more sub-stantive logical properties of belief, it is necessary to relate beliefs at dif-ferent states of the world to each other and to the truth. Thus positiveintrospection requires that if something would be believed at every statethat I think possible, then I must believe it. Negative introspection requiresthat if something would not be believed at every state that I believepossible, then I must not believe it. The knowledge axiom requires that ifI believe something, it must be true. These three properties together havebeen shown to be equivalent to economists' standard assumption of aninformation partition.

The assumption of an information partition is necessary as long as thedecision maker ``understands'' the structure of the state space, so that weassume that part of his background knowledge is that he knows what hewould have known at each state of the world. But there is no need to makethat assumption and a number of papers in the economics literature haveconsidered models where a decision maker's information is not representedby an information partition.2 While the negative introspection assumption,

2 STEPHEN MORRIS

2 Geanakoplos [10], Rubinstein and Wolinsky [23], Samet [24], Shin [26],Brandenburger et al. [5], Bacharach [3].


say, can be objected to on various grounds, a weakness of this literatureis that there is no decision theoretic foundation for alternative assump-tions.

In order to relate these more substantive logical properties of belief toproperties of preferences, it is necessary to examine how preferences atdifferent states of the world are related to one another. The preferences ateach state of the world should somehow reflect the fact that it is the samedecision maker at each state of the world. There should be a sense in whichthe decision maker as choices at each state of the world reflect some ``meta-ordering,'' which represents that decision maker's preferences independentof the state of the world. Thus consider the set of ``acts'' associating aconsequence with each state of the world. Preferences are said to becoherent if there exists some meta-ordering over acts with the followingproperty. Consider a decision problem��that is, a finite set of acts availableto the decision maker. Suppose that at each state |, the decision makermakes an optimal choice according to his state | preferences. An outsideobserver could calculate the consequence in each state of such optimalchoices, which generates a new act. For coherence, it is required that thatact is at least as good, under the meta-ordering, as any act which wasavailable in the original decision problem. It will be shown that if preferencesare coherent, then belief satisfies the knowledge axiom and positive intro-spection but need not satisfy negative introspection. It thus accords with astandard weakening of partition information in the philosophy andcomputer science literatures.

From the viewpoint of an outside observer, coherence is a normativerestriction. When would the decision maker be better off by making choiceswhich depend on the state via preferences at each state? ``Better off '' ismeasured in ex ante terms, by the meta-ordering, relative to a constant(not state contingent) choice under the meta-ordering. But this normativeinterpretation must be with respect to an outside observer with access tothe meta-ordering and an understanding of the state space not available tothe decision maker. In the conclusion, I discuss why coherence is importantin economic applications.

This analysis is extended to allow beliefs to vary through time. Say thatbeliefs satisfy valuable information if, in addition to being coherent, choicesmade at later dates always make the decision maker (weakly) better offunder the meta-ordering. This property, which is the natural dynamicanalog of coherence, implies refinement: if something is believed at somedate, then it is still believed at all future dates. But it turns out to requiremore. As long as beliefs do not change, negative introspection is notnecessary. But once beliefs change, it must be the case not only that youbelieve more as time goes on, but also that you believe that you did notbelieve those things you did not believe in the past.

3THE LOGIC OF BELIEF


Thus there are three main results of the paper, each giving necessaryconditions on beliefs given that, respectively, preferences are normal,coherent, and satisfy valuable information. Each of these results is tight:examples in the paper show that stronger conditions are not necessary.

This work extends a line of research started by Geanakoplos [10] andfollowed by Morris and Shin [21]. Geanakoplos considered the impact ofweakening the assumption of partition information in economics. Assumingexpected utility maximization and Bayes updating, Geanakoplos showedthat the knowledge axiom and positive introspection were necessary forinformation to be valuable, and that negative introspection was notnecessary.3 Morris and Shin derived analogous results for belief revisionand showed that if Bayes updating is not imposed, the knowledge axiomand positive introspection are also sufficient for information to be valuable.

This paper extends that work in two major ways. First of all, expectedutility maximization need not be assumed for the results. The increasedgenerality of the results is useful but less interesting than the new methodsand perspective which dropping expected utility implies. We derivecompletely different kinds of proofs (not relying on the linear algebra of theexpected utility proofs), and the more abstract interpretation of the resultsoutlined above. The valuable information condition appears in a staticsetting as the coherence condition, and is thus reinterpreted as a restrictionon preferences.

Secondly, the results in this paper are fully dynamic and provide anintriguing twist on the earlier work. When we generalize the valuable infor-mation condition to a dynamic environment, we partially reverse theconclusion from the earlier work that negative introspection is notnecessary for valuable information. It is true that it is not necessary for thiscondition to hold contemporaneously. But it turns out to be necessary thatif something was not believed in the past, then it must be believed now thatit was not believed. In this sense, the departure from the assumption ofpartition information is small.

The paper is organized as follows. Section 2 sets up the framework formodeling and relating preferences and beliefs. Normal belief systems areshown to correspond to preference relations which are complete orderings.Section 3 introduces more substantive properties of belief and shows thatthe knowledge axiom and positive introspection are necessary forcoherence. Section 4 presents a dynamic model of belief and shows thatrefinement and historical negative introspection are necessary for valuableinformation. Section 5 concludes.

4 STEPHEN MORRIS

3 The knowledge axiom, positive introspection, and a third property called ``nestedness''were shown to be jointly necessary and sufficient.


2. Static Preferences and Beliefs (Foundations)

2.1. Beliefs

Consider a fixed finite set 0 of ``possible worlds'' or states. Take asa primitive representation of beliefs an ``accessibility'' or ``possibility''relation: write |$ # P(|) if state |$ is believed possible when the true stateis |. This way of modeling qualitative belief originates in the Kripkesemantics for modal logic (Kripke [14]) as developed by Hintikka [13]for epistemic logic. ``Belief '' is used here in a non-probabilistic sense: it isanalogous to ``belief with probability one'' in probabilistic models.4

Now P(|) is the set of states considered possible when the true state is|. An event is then believed if it contains every state considered possible.This suggests the following formal definition of belief.

Definition 1. The operator B: 20 � 20 represents the possibility relationP if

B(E)=[|: P(|)/E].

Thus if operator B represents a decision maker's possibility relation, weshall say that the decision maker believes event E at state | if and only if| # B(E).

Definition 2. The operator B is a normal belief operator if there existsa relation P such that B represents P.

The term normal is intended to draw a parallel with normal systems ofmodal logic in which the logic is sound and complete with respect to someversion of the Kripke semantics. The following theorem is well known inother contexts (see Chellas [6]) but is repeated and proved in thisframework for completeness.

Theorem 1. Belief operator B is a normal belief operator if and only ifB satisfies

B1 (distributivity): B(E & F )=B(E) & B(F ), for all events E, F.

B2 (belief in tautologies): B(0)=0.

The distributivity property states that event E is believed and event F isbelieved if and only if the event E & F is believed. The tautology propertystates that the universal event 0 is always believed. The theorem states that


4 In the economics literature, the term ``knowledge'' is often used synonymously with ``beliefwith probability one.'' Because beliefs may be false in the framework I present, I do not followthis usage.


if operator B represents a possibility relation (i.e., is a normal beliefoperator) then it satisfies these two properties. Conversely, if an operatorB satisfies these two properties, then there exists a possibility relationwhich generates the operator. In fact, the following proof makes clear thatthat possibility relation is unique.

Proof of Theorem 1. If B represents P, then B(E & F )=[|: P(|)/E & F]=[|: P(|)/E] & [|: P(|)/F]=B(E) & B(F); B(0)=[|: P(|)/0]=0. Now suppose B satisfies B1 and B2. B1 implies the followingmonotonicity property:

E/F O B(E)/B(F ), for all events E, F. (2.1)

Write &E for the complement of E in 0 and define P by

P(|)=[|$: | � B(&[|$])]. (2.2)

I will show that B represents the P constructed from it by (2.2). Suppose| # B(E). Then E/ &[|$] implies (by 2.1) | # B(&[|$]) implies |$ �P(|). Thus |$ # P(|) implies E/3 [&|$] implies |$ # E. Thus P(|)/E.

Conversely, suppose P(|)/E. If E=0, | # B(E) by B2. Suppose thenthat E{0. Then

E= ,|$ � E

&[|$]. (2.3)

But | � B(&[|$]) implies |$ # E; so |$ � E implies | # B(&[|$]). Thus| # B(&[|$]) for all |$ � E implies (by 2.3) | # B(E). K

The characterization of (2.2) is of independent interest: state |$ isthought possible at state | if and only if the complement of |$ in 0 is notbelieved.

2.2. Preferences

I now describe a decision maker's preferences. The finite state space 0 isgiven. As in Savage [25], an individual has preferences over ``acts,'' wherean act determines a certain known outcome depending on which state ofthe world is realized. For simplicity, I restrict the set of outcomes to thereal line, so that we can think of the decision maker receiving a moneyprize. Thus act x # R0 gives a prize x| in state | # 0, where x| , is the | thcoordinate of x. For event E/0, xE denotes the tuple [x|]| # E . Thus ifz=(xE , y&E), z|=x| , for all | # E, and z|=y| , for all | � E. In thisnotation, x| and x[|] are equivalent; similarly, x&| and x&[|] are usedinterchangeably. For any number c, write c for the act giving c in everystate. Thus 0 is the act which gives 0 in every state.

6 STEPHEN MORRIS


At each state of the world, | # 0, the decision maker is assumed to havea preference relation, p| , over acts, with the interpretation that xp| ymeans that act x is at least as good as act y for the decision maker if thetrue state is |.

This approach to modeling preferences is quite novel and it will be usefulto contrast it with two apparently related but distinct issues. A preferencerelation over acts is said to have a state dependent representation if theutility derived from a given outcome varies with the state. The preferencerelation p| , may or may not be state dependent in that sense (in exampleslater, we will for convenience assume that it is state independent). The |subscript in xp| y refers only to the fact that the decision maker wouldchoose act x to act y if the true state was |. It has nothing to do with whathis preferences would be if he knew that the true state was |.

Our usage also should not be confused with conditional preferences.Suppose a decision maker had preference relation p* over acts. If he wasinformed that event E was true, one can think of various rules for deter-mining what his conditional preferences would�should be. Our approach isessentially the other way round. At state |, a decision maker's preferencesare represented by p| . Any beliefs or knowledge of the decision maker arereflected in his preferences. Given these preference relations, we will laterask whether there is some meta-ordering of acts, p*, which the preferencerelations [ p|]| # 0 somehow reflect.

Definition 3. Relation p is completely ordered if it satisfies, for allx, y, z # R0,

P1 (completeness): xpy or ypxP2 (transitivity): xpy and ypz O xpz.

Preferences relations [ p|]| # 0 are completely ordered if relation p| iscompletely ordered for each | # 0. Strict preference relations and indif-ference relations are defined in the usual way by xo| y if xp| y andyp� | x and xt| y if xp| y and yp| x.

2.3. Beliefs and Preferenes

It is natural to define belief in terms of preferences as follows. If the deci-sion maker's preferences never depend on anything that happens whenevent E does not occur, then the decision maker believes E. On the otherhand, if the decision maker is ever concerned about what happens when Edoes not occur, then he cannot believe E.

Definition 4. Belief operator B reflects preference relations [ p|]| # 0 if

B(E)=[|: (xE , y&E]t| (xE , z&E) for all x, y, z # R0.



Savage [25] informally defined knowledge in exactly this way. Thus adecision maker believes an event E (at |) if the complement of E is null(under p|) in Savage's sense. Notice that any given belief operator reflectsmany preference relations. When no confusion arises, I will leave implicitthe preference relations which a belief operator reflects.

I will be concerned how properties of preference relations are related toproperties of belief operators. I first show how the standard axioms onpreferences assumed above translate into standard logical properties ofbelief.

Theorem 2. If preference relations are completely ordered, then thebelief operator representing those preference relations is normal.

It is also possible to show a converse result. For any normal beliefoperator B, there exist completely ordered expected utility preference rela-tions such that B reflects those preference relations.

Proof of Theorem 2. (B1) (i) Suppose | # B(E) & B(F ). | # B(E) O(xE & F , y&(E & F ))p| (xE & F , yE & &F , z&E) for all x, y, z # R0. | # B(F ) O(xE & F , yE & &F , z&E)p| (xE & F , z&(E & F)) for all x, y, z # R0. Thus bytransitivity, (xE & F , y&(E & F ))p| (xE & F , z&(E & F )) for all x, y, z # R0. So| # B(E & F ).

(ii) Suppose | # B(E & F ) . (xE & F , y&(E & F ))p| (xE & F , z&(E & F)) forall x, y, z # R0. Thus (xE , y&E)p| (xE , z&E) and (xF , y&F)p| (xF , z&F)for all x, y, z # R0. So | # B(E) & B(F ).

(B2) By completeness, xp| x, for all x # R0, | # 0. Thus(x0 , y&0)p| (x0 , z&0), for all x, y, z # R0, | # 0. So | # B(0) for all| # 0, i.e., B(0)=0. K

By Theorem 1, any belief operator which reflects completely orderedpreferences represents a possibility relation. Equation (2.2) in Proof ofTheorem 1 can be used to show that this possibility relation is

P(|)=[|$: (x|$ , z&|$)o|( y|$ , z&|$) for some x, y, z # R0]. (2.4)

2.4. Expected Utility Preferences

One objective of this paper is to study the relation between belief andpreferences without putting too much structure on preferences. In par-ticular, the preference relation p| may or may not have an expected utilityrepresentation. But imposing standard axioms��such as the sure thingprinciple��on each relation p| would ensure an expected utility represen-tation.5 For simplicity, we will impose some additional structure. Suppose

8 STEPHEN MORRIS

5 Davidson and Suppes [7] and, more recently, Gul [12] provide axiomatizations forsubjective expected utility in a finite state space setting.


that the decision maker's utility function over outcomes depends neitheron the true state nor on the state in which the outcome occurs. At eachstate |, he has some probability distribution over possible states. Thus atstate |, he believes that the true state is |$ with probability $(|$ | |). Thiscan be represented formally as follows.

Definition 5. Preference relations [ p|]| # 0 have a (state independent)expected utility representation if there exists a strictly increasing andcontinuous utility function u: R � R and, for each state | # 0, a credencefunction $( } | |): 0 � R+ , with �|$ # 0 $(|$ | |)=1, such that

xp| y � :|$ # 0

$(|$ | |) u(x|$)� :|$ # 0

$(|$ | |) u( y|$).

In this case, the belief operator and possibility relation defined above areextremely natural. An event is believed if and only if it is assigned probabilityone, i.e.,

B(E)={|: :|$ # E

$(|$ | |)=1= .

A state is considered possible if it is assigned strictly positive probability,i.e.,

P(|)=[|$: $(|$ | |)>0].

2.2. Monotonicity and Continuity

The results of this paper remain correct, exactly as stated, if the expectedutility assumption is made throughout. But the assumption is not madesince the results do not rely on it, and the intuition is in some ways moretransparent without it. In this section, however, I introduce regularityconditions on preferences (which are implied by, but strictly weaker than,expected utility maximization) which simplify the analysis. An earlierversion of this paper (Morris [19]) derived analogous results to those ofthis paper, without making these assumptions.6

Write x�y if x|�y| for all | # 0, x>y if x�y and x|>y for some| # 0; and xry if x|>y| for all | # 0. The following properties of apreference relation p will be assumed.


6 The main role of these assumptions is to ensure (via Lemma 1) that there is no ambiguityin the definition of belief. Morris [20] considers alternative notions of belief without theserestrictions.


P3 (continuity): The set [x # R0|xpy] is closed for all y # R0.

P4 (monotonicity)7 : (i) If x�y, then xpy. (ii) If (x| , z&|)p

( y| , z&|) for some x, y, z # R0, then (x| , z&|)o ( y| , z&|) for allx, y, z # R0 such that x|>y| . (iii) If xry, then xoy.

These assumptions allow us to give an alternative characterization of thepossibility relation P. The earlier characterization (Eq. (2.4)) showed thata state |$ was considered possible if the decision maker ever took intoaccount what happened at state |$. The characterization of the followinglemma shows that a state |$ is considered possible if the decision maker isalways better off if he gets more at state |$, even if he gets = less at all otherstates, for = sufficiently small.

Lemma 1. If preferences relations [ p|]| # 0 satisfy P1 through P4, then

P(|)=[|$: for all xry, there exists zRy such that (x|$ , z&|$)o| y].

Proof of Lemma 1. ( O ) |$ # P(|) implies (by P4(ii)) (x|$ , y&|$)o| yfor all x, y # R0 with x|$>y|$ . By P3, there exists zRy such thatx|$ , z&|$)o| y.

( o ) suppose for some xryrz, (x|$ , z&|$)o| y; then (by P4(i))(x$ , y&|$)p| (x|$ , z&|$)o| y and so |$ # P(|). K

In the remainder of the paper, preferences are always assumed to satisfyP1 through P4.

3. Further Static Properties of Preferences and Beliefs

In the previous section, a decision maker was assumed to have preferencesat each state of the world. Belief was defined as a property of thosepreferences. Various minimal properties of those preferences were related tocertain minimal properties of belief. In this section, more fundamentalproperties of preferences and belief will be introduced and related together.

3.1. Further Properties of Belief

I first introduce the properties of belief operators which will be critical.This treatment of beliefs and knowledge was introduced into the economicsliterature by Milgrom [16] and Bacharach [2]; more detailed discussionof the following material is available in the surveys of Aumann [1],Bacharach [3], Binmore and Brandenburger [4], and Geanakoplos [11].

10 STEPHEN MORRIS

7 There is some redundancy in this definition. Note that (ii) implies (i), and that (ii) implies(iii) as long as xoy for some x, y.


A first restriction on belief is the requirement that the decision maker hasaccess to his own beliefs, and thus if he believes something, he believes thathe believes it.

B3 ( positive introspection8): B(E)/B(B(E)), for all events E.

Beliefs are correct if whenever an event E is believed, it is in fact true.Knowledge is usually defined to be correct belief.

B4 (knowledge axiom9): B(E)/E, for all events E.

Shin [26] showed that a decision maker's beliefs satisfy properties B1through B4 if and only if belief is equivalent to a formal notion ofprovability. This system of beliefs��known as (S4) in the logicliterature��strikes many commentators as excessively strong, leading tomany attempts to weaken various of the axioms. However, it does notimply the following important additional property of belief.

B5 (negative introspection10): &B(E)/B(&B(E)), for all events E.

One interpretation is that B5 allows the decision maker to make deductionsabout what state he is in from what he would have believed if he were notin the state he is in. Properties B1 through B5 can be translated intoproperties of the equivalent possibility relation.

Lemma 2. Belief satisfies B3 if and only if the possibility relation istransitive, with |" # P(|$) and |$ # P(|) O |" # P(|), for all |", |$, | # 0(equivalently, |$ # P(|) O P(|$)/P(|), for all |$, | # 0). Belief satisfiesB4 if and only if the possibility relation is reflexive, with | # P(|) for all| # 0. Belief satisfies B5 if and only if the possibility relation is euclidean,with |" # P(|) and |$ # P(|) O |" # P(|$), for all |", |$, | # 0 (equiv-alently, |$ # P(|) O P(|)/P(|$), for all |$, | # 0).

Discussion and proofs of Lemma 2 are available in the papers cited atthe beginning of this section.11 Lemma 2 shows that assumptions B1through B5 imply that the decision maker's beliefs can be representedby a partition. To see why, note that B3 and B5 jointly implyP(|)/[|$ # 0 | P(|$)=P(|)], while B4 implies that [|$ # 0 | P(|$)=P(|)]/P(|). So B3 through B5 imply P(|)=[|$ # 0 | P(|$)=P(|)] forall | # 0.

Some examples will illustrate these properties.


8 This name is used by Fagin et al. [9]. Hintikka [13] called this knowing that one knows.9 Geanakoplos [10] called this property non-delusion.10 Fagin et al. [9] used this name. Bacharach [3] calls this the axiom of wisdom.

Geanakoplos [10] called it knowing that you don't know.11 The proof of Lemma 5 in section 4 gives a proof of the possibility relation version of B5.


Example 1.

0=[a, b, c]

P(a)=[a, b]

P(b)=[b]

P(c)=[b, c]

Here belief satisfies positive introspection and the knowledge axiom butfails to satisfy negative introspection, since b # P(a) and P(a)/3 P(b). Thusthe event [b] is not believed at states a and c, i.e, &B([b])=[a, c]; butthat event is never believed, i.e., B(&B([b]))=B([a, c])=<.

Example 2.

0=[a, b, c]

P(a)=[a, b]

P(b)=[b]

P(c)=[a, c]

Here belief satisfies the knowledge axiom but fails to satisfy positiveintrospection, since a # P(c) and P(a)/3 P(c). Thus the event [a, c] isbelieved only at state c, i.e., B([a, c])=[c]; but that event is neverbelieved, i.e., B(B([a, c]))=B([c])=<. Negative introspection also failsin this example.12

3.2. Equivalent Properties of Preferences

In the next section, I will show that a certain coherence restriction onpreferences is equivalent to properties B1 through B4. In this section, I firstgive a direct interpretation of the positive introspection axiom (P3) and theknowledge axiom (P4) in terms of preferences.

P5 (non-triviality): (x| , z&|)o| ( y| , z&|) for some x, y, z # R0, forall | # 0.

Non-triviality requires that preferences at state | are sensitive to whathappens at state |.

Lemma 3. If preference relations satisfy non-triviality P5, then beliefssatisfy the knowledge axiom B4.

12 STEPHEN MORRIS

12 In fact, negative introspection and the knowledge axiom imply positive introspection.


This lemma is simply a restatement of the relevant definitions.Given preference relations [ p|]| # 0 and event E, say that xpE y if

xp| y for all | # E; and xoE y if xpE y and xo| y for some | # E.

P6 (extended sympathy13): xo P(|) y implies xo| y, for all x, y #R0, | # 0.

Extended sympathy requires that preferences at a given state are sen-sitive (in the most minimal way) to what preferences would have been atany state that is considered possible.

Lemma 4. If preference relations satisfy extended sympathy P6, thenbeliefs satisfy positive introspection B3.

Proof of Lemma 4. I prove the contrapositive. If (B3) fails then (byLemma 2) there exists a, b, c # 0 such that a # P(b), b # P(c), but a � P(c).Now let xa=1 and x|=0 for all |{a. By P4(ii), xob 0 and by P4(i),xp| 0 for all | # 0. Thus xoP(c) 0. But a � P(c) implies 0tc x, since x and0 agree at all states other than a. But xoP(c) 0 but xtc 0 contradictsextended sympathy.

Properties P5 and P6 were direct restatements of the equivalent beliefproperties. In the next section, a more subtle characterization is given.

3.3. Coherence

The coherence property introduced in this section is intended to be aminimal rationality requirement relating together preferences at differentstates of the world. Is it the case that the choices made at different statesof the world can be seen as reflecting a true, metapreference ordering overacts? Let us first define what such a meta-preference ordering would looklike.

Definition 6. A preference relation p is a meta-ordering if it iscomplete (P1), transitive (P2), continuous (P3), and satisfies the followingstrong monotonicity condition, for all x, y, z # R0:

(x| , z&|)p ( y| , z&|) � x|�y| .

I want to make a comparison between what would happen if a decisionmaker had such a meta-ordering and had to make a constant (not statecontingent) choice in a decision problem and what would happen if hewas able to make, at each state, a choice which was optimal given his


13 This name and property was suggested by a referee.


preferences at that state (in the same decision problem). Is it the case thatthe decision maker is made no worse off (in terms of his meta-ordering) bybeing allowed to make state contingent choices? If this is always true (forevery decision problem) then his preferences are said to be coherent. Tostate this formally, additional notation is required.

A decision problem is a finite subset of acts, D/R0. For each decisionproblem, we can define, for each state |, the set of optimal choices,C|[D]=[x # D: xp| y for all y # D]. By the finiteness of D, andcompleteness and transitivity of each p| , these optimal choice sets arenon-empty for every decision problem.

Now what will happen if the individual makes an optimal choice at eachstate of the world? A decision rule is a function f: 0 � D and an optimaldecision rule satisfies f (|) # C|[D] for all | # 0. Note that the optimalityis ex post, so that the decision is optimal with respect to (ex post) beliefsat the actual state. When such an optimal decision rule is followed, f (|)determines what happens only at state | (and not at other states thoughtpossible at |). So such an optimal decision rule generates a new act y, notnecessarily in D, with y|=f|(|), for all | # 0, (where f|$(|) is the |$thcoordinate of f (|)). Thus the class of acts generated by optimal decisionrules is C*[D]=[x # R0: x|=f| (|) , for all | # 0, for some optimal f ].A minimal coherence condition on preferences is that acts in C*[D](which may or may not be contained in D) must be at least as good (undersome meta-ordering p*) as acts in the original decision problem, D, i.e.,xp* y for all D, x # C*[D] and y # D. It will be convenient to use aslightly weaker condition:14

P7 (coherence): There exists a meta-ordering p* such that for eachfinite D/R0, there exists x # C*[D] such that xp* y, for all y # D.

Theorem 3. If preference relations are coherent, then the beliefs reflectingthose preference relations satisfy the knowledge axiom and positive intro-spection.

Before proving the theorem, it is useful to consider some illustrativeexamples. The first exhibits some coherent preferences.

Example 3. Suppose 0=[a, b, c] and preference relations have thefollowing representation:

14 STEPHEN MORRIS

14 For generic decision problems, there will always be a unique optimal choice, so there isno difference between the conditions. All the results in this paper would go through (withslightly different arguments) if the stronger version was used. Morris [18], Section 2, has adetailed investigation, in a related context, of different versions of this axiom.


xpa y � 12 xa+ 1

2 xb� 12 ya+ 1

2 yb

xpb y � xb�yb

xpc y � 12 xb+ 1

2 xc� 12 yb+ 1

2 yc .

Thus the decision maker is a risk neutral expected utility maximizer,with the possibility relation of Example 1 (which satisfied the knowledgeaxiom and positive introspection). Define a meta-ordering p* by:

xp* y � 16 xa+ 2

3 xb+ 16 xc� 1

6 ya+ 23 yb+ 1

6 yc .

To show that coherence is satisfied with this meta-ordering, consider anydecision problem D and optimal decision rule f. Then, for any y # D,optimality of f (a) implies

12 fa(a)+ 1

2 fb(a)� 12 ya+ 1

2 yb . (3.1)

Optimality of f (b) implies fb(b)�yb (3.2); fb(b)�fb(a) (3.3), andfb(b)�fb(c) (3.4). Optimality of f (c) implies

12 fb(c)+ 1

2 fc(c)� 12 yb+ 1

2 yc . (3.5)

But multiplying Eqs. (3.1), (3.2), and (3.5) by 1�3, Eqs. (3.3) and (3.4) by1�6, and summing gives the equation

16 fa(a)+ 2

3 fb(b)+ 16 fc(c)� 1

6 ya+ 23 yb+ 1

6 yc .

Thus xp* y for all x # C*[D]. But since this holds for each y # D,coherence holds.

The logic of Example 3 generalizes, and it can be shown that for anybelief operator satisfying the knowledge axiom and positive introspection,it is possible to construct coherent preferences such that the belief operatorreflects those preferences (this is proved in an earlier version of the paper,Morris [19]).

The following example illustrates why the knowledge axiom is necessary.

Example 4. Suppose 0=[a, b] and preference relations have thefollowing representation:

xpa y � xa�ya

xpb y � xa�ya .



Thus P(a)=P(b)=[a]. The knowledge axiom fails but positive introspec-tion holds. Now consider decision problem D=[0, x] where xa== andxb=&1. For = sufficiently small, 0o* x. But C*[D]=[x]. So coherencefails.

The following example illustrates why positive introspection is necessaryfor coherence.

Example 5. Suppose preferences have the following representation, forsome :, ; # (0, 1):

xpa y � :xa+(1&:) xb�:ya+(1&:) yb

xpb y � xb�yb

xpc y � ;xa+(1&;) xc�;ya+(1&;) yc .

The decision maker is thus a risk neutral expected utility maximizer withthe possibility relation of Example 2 (so beliefs satisfy the knowledge axiombut positive introspection fails). Consider the decision problem D=[x, 0],where xa==, xb=&1, and xc=&=2. Now, whatever the values of : and;, 0 is optimal at b. For all = sufficiently small but positive, 0 is optimal ata and x is optimal at c. Thus C*[D]=[(0a , 0b , xc)]=[(0, 0, &=2)]. But0=(0, 0, 0)p* (0, 0, &=2), for any meta-ordering p*. So coherence fails.

Note that this argument is true for any : and ;, and thus for any riskneutral expected utility maximizer with the possibility relation of Example2. The following proof of Theorem 3 generalizes the logic of Examples 4and 5 to non-risk neutral non-expected utility preferences.

Proof of Theorem 3. Suppose there exists p* satisfying the coherenceproperty, but the knowledge axiom B4 fails. Then there exists a # 0 witha � P(a). By continuity, it is possible to choose =>0 such that xo* 0,where xa=1 and x|=&= for all |{a. Consider the decision problemD=[x, 0]. Now a � P(a) implies &=pa x; monotonicity (P4(iii)), implies0oa &=; and so transitivity (P2) implies 0oa x. Thus Ca[D]=[(0)] andya=0 for all y # C*[D]. Now monotonicity (P4(i)) implies xo* 0p* yfor all y # C*[D]. So coherence fails. Now suppose there exists p* satisfy-ing the coherence property, the knowledge axiom (B4) holds but positiveintrospection (B3) fails. By the characterization of Lemma 2, there exista, b, c # 0 such that b # P(a), a # P(c), but b � P(c); by B4, b # P(b). Bycontinuity, it is possible to choose =1 and =2 such that 0oa x, 0ob x, andxoc0 where xa==1 , xb=&1, xc=&=2 , and x|=0 for all | � [a, b, c].Thus y # C*[D] implies yc=&=2 and y|=0 for all |{c. So 0o* y for ally # C*[D], contradicting coherence. K

16 STEPHEN MORRIS


4. Dynamic Preferences and Belief

Now extend preferences to many time periods. Say there are a finitenumber of time periods, t=0, 1, ..., T. The individual's preferences in state| in time period t are represented by a relation on R0, p|, t , satisfyingP1�P4. A belief operator, Bt , at each date t is now naturally defined by

Bt(E)=[|: (xE , y&E)t|, t(xE , z&E), for all x, y, z # R0].

I write Pt for the possibility relation which Bt represents. Now I will saythat beliefs satisfy a certain property of belief if each Bt satisfies it. Butfurther properties of belief dealing with how belief changes through timewill also be needed.

B6 (refinement): Bs(E)/Bt(E), for all events E and s�t.

Belief satisfies refinement if you never revise your beliefs, so thatanything believed at date s is still believed at any later date t. Thus beliefonly expands. This represents a very conservative notion of belief in whichnothing is ever believed which will later not to be believed. Notice thatbelief satisfies refinement only if the equivalent possibility relations satisfyPt(|)/Ps(|), for all | # 0, s�t.

The following dynamic restriction on preferences requires not only thecoherence property of the previous section, but also that decisions made ata later date are more valuable (in terms of the meta-ordering) thandecisions made at an earlier date. The approach here involves fixing the setof possible decisions, D, through time and comparing choices. One couldalso study changing decision problems, but the focus here is on changingpreferences and beliefs, for given decision problems.

Generalize optimal choices and optimal profiles in the natural way:C|, t[D]=[x # D: xp|, t y, for all y # D]. A decision rule, f: 0 � D, isoptimal at time t if f (|) # C|, t[D] for all | # 0. Ct*[D]=[x # R0:x|=f|(|), for all | # 0, for some f optimal at time t].

P8 (valuable information): There exists a meta-ordering p* suchthat for each finite D/R0, and each t=1, ..., T, there exists x(t) # Ct*[D]such that

x(t)p* y, for all y # D & C0*[D] & } } } & C*t&1[D].

As with coherence, if the decision maker knew enough to check thiscondition, he would presumably revise his preferences. But suppose therewas a meta-ordering p* which represented the ``true interests'' of thedecision maker in a poorly understood world. What rules for revisingpreferences and thus beliefs would serve him well? Valuable information



offers one minimal criterion for such rules: they should never make himworse off.

A natural conjecture might be that refinement B6 together with theearlier requirements of positive introspection B3 and the knowledge axiomB4 would be the only necessary conditions for valuable information P8.The following example shows that this is not the case. Assume for sim-plicity that preferences are risk neutral expected utility preferences (Proofof Theorem 4 generalizes the argument to arbitrary preferences). Supposethat the belief operators reflecting preferences represent the followingpossibility relations.

Example 6. 0=[a, b, c]

P0(a)=[a, b, c] P1(a)=[a, b, c] P2(a)=[a, b, c]

P0(b)=[a, b, c] P1(b)=[a, b, c] P2(b)=[b, c]

P0(c)=[a, b, c] P1(c)=[c] P2(c)=[c]

Belief satisfies positive introspection, the knowledge axiom, and refine-ment. But in the following decision problem, valuable information fails.Consider the decision problem D=[0, x], where xa= &1, xb=&=2, andxc==. Now for any given beliefs, for =>0 sufficiently small, act x will bestrictly optimal with possibility set [b, c] and act 0 will be strictly optimalwhenever the possibility set is [a, b, c]. Act x will also be strictly optimalwhenever the possibility set is [c]. So C2*[D]=[(0, &=2, =)], whileC1*[D]=[(0, 0, =)]. So y # B1*[D] and z # B2*[D] implies yo* z,contradicting valuable information.

What goes wrong in this example? It is true that negative introspectionfails. But negative introspection cannot be necessary for valuable informa-tion. Consider the case where preferences stay constant through time. Inthis case, the valuable information condition reduces to coherence, andExample 3 has already shown that negative introspection is not necessaryfor coherence.

It turns out that a subtle weakening of negative introspection is required.If, in state | at date s, you don't believe E, and if at state | at date t>s,you believe something more than you did at date s, then you must in par-ticular believe that you didn't believe E at date s. This property will belabelled historical negative introspection. In defining it formally, the followinglemmas will be useful.

Lemma 5. &Bs(E)/Bt(&Bs(E)) for all events E if and only if|$ # Pt(|) O Ps(|)/Ps(|$) for all |, |$ # 0.

18 STEPHEN MORRIS


Proof of Lemma 5. The following statements are equivalent.

&Bs(E)/Bt(&Bs(E)), for all E.

Ps(|)/3 E implies Pt(|)/&Bs(E), for all |, E.

|$ # Pt(|) and Ps(|)/3 E imply Ps(|$)/3 E, for all |, |$, E.

|$ # Pt(|) implies Ps(w)/Ps(|$), for all |, |$. K

Lemma 6. Suppose beliefs satisfy refinement (B6) and s<t. Then | #Bt(E) & &Bs(E), for some event E, if and only if Ps(|){Pt(|).

Proof of Lemma 6. Suppose Ps(|){Pt(|)=E. By definition,| # Bt(E), while by refinement, | # &Bs(E). Suppose | # Bt(E) & &Bs(E)for some E. Then Pt(|)/E and Ps(|)/3 E and so Ps(|){Pt(|). K

Historical negative introspection requires that the property of Lemma 5holds exactly at those states where the property of Lemma 6 holds.

B7 (historical negative introspection): &Bs(E) & Bt(F ) & &Bs(F )/Bt(&Bs(E)), for all events E and F, s�t.

Let us compare this property with negative introspection. Suppose thatyou don't believe something at time s. Negative introspection requires thatyou believe that you don't believe it at time s. If beliefs satisfy refinement,then you will continue believe (in all future periods) that you didn't believeit. Thus if didn't believe something at time s, you believe that you didn'tbelieve it from time s on. Historical negative introspection requires thissame property (if you didn't believe something, you believe that you didn'tbelieve it...), but this property must hold only once you have learnedsomething you didn't know at time s. Thus as long as your beliefs remainconstant, historical negative introspection is vacuous. But if you believesomething new at every date, then historical negative introspection isequivalent to negative introspection with a one period lag. In this sense,historical negative introspection is a small weakening of negative intro-spection.

The following characterization of historical negative introspection is usedin the proof.

Lemma 7. Suppose beliefs satisfy refinement (B6). Then beliefs satisfyhistorical negative introspection if and only if |$ # Pt(|){Ps(|) impliesPs(|)/Ps(|$), for all |$, | # 0, s�t.

Proof of Lemma 7. Follows immediately from Definitions and Lemmas5 and 6. K

It is now possible to give the main result of this section.



Theorem 4. If preference relations satisfy valuable information, then thebeliefs reflecting those preference relations satisfy the knowledge axiom,positive introspection, refinement, and historical negative introspection.

The proof of the theorem is a generalization of the logic of Example 6.

Proof of Theorem 4. (1) Valuable information implies coherence (bydefinition) which implies the knowledge axiom and positive introspection(by Theorem 3).

(2) Suppose valuable information, the knowledge axiom, andpositive introspection hold but refinement fails. Then there existss<t, a # Pt(b), and a � Ps(b). Observe that (i) by positive introspection,| # Ps(b) O Ps(|)/Ps(b); (ii) by the knowledge axiom, | � Ps(b) OPs(|) & &Ps(b){< (since | # Ps(|) & &Ps(b)); (iii) by assumption,Pt(b) & &Ps(b){<. Let x|=&= if | # Ps(b) and x|=1 otherwise. Bycontinuity, it is possible to choose = such that, by (i), 0o| x for all| # Ps(b), by (ii), xo| 0 for all | � Ps(b), and by (iii), xob 0. LetD=[0, x]. Now y # Cs*[D] O y|=0 for all | # Ps(b) and y|=1 for all| � Ps(b). But z # Ct*[D] O yb=&=, y|�0 for all | # Ps(b) and y|�1 forall | � Ps(b). So yo* z for all y # Cs*[D] and z # Ct*[D], contradicting theassumption of valuable information.

(3) Suppose valuable information, the knowledge axiom, refinement,and valuable information hold, but historical negative introspection fails.Thus there exist s<t, b # Pt(a){Ps(a) with Ps(a)/3 Ps(b). Let A=[|: Pt(|)=Pt(a)]; B=[|: Pt(|)/Pt(b)]; C=&Pt(a). Note thatB=Pt(b) (by knowledge axiom and positive introspection), A & C=<,and B & C=< by definition. Let us show that A & B=<.

Suppose | # A & B; | # A O Pt(|)=Pt(a); | # B O Pt(|)/Pt(b). Thusa # Pt(a)/Pt(b) by the knowledge axiom; a # Ps(b) by refinement;Ps(a)/Ps(b) by positive introspection, contradicting assumption.

Thus A, B, and C are mutually disjoint. The following properties will beuseful:

(i) Ps(|) & A{< for all | # A [since | # Ps(|) & A].

(ii) Ps(|) & C{< for all | # A [since Pt(|)=Pt(a), Ps(|)=Ps(a), C=&Pt(a), Pt(a)/Ps(a), and Pt(a){Ps(a)].

(iii) Pt(|) & B{< for all | # A [since b # Pt(|) & B].

(iv) Pt(|) & C=< for all | # A [since C=&Pt(a)=&Pt(|)].

(v) Ps(|) & A=< for all | # B[| # B O | # Pt(b) O (by B6)| # Ps(b) O (by B3) Ps(|)/Ps(b). Suppose |$ # Ps(|). Then |$ # Ps(b) OPs(|$)/Ps(b). Suppose also |$ # A. Then Pt(|$)=Pt(a) O (by B3 and B6

20 STEPHEN MORRIS


Ps(|$)=Ps(a) Thus Ps(a)/Ps(b), contradicting our assumption. Thusthere does not exist such an |$.].

(vi) Ps(|) & B{< for all | # B [since | # Ps(|) & B].

(vii) Ps(|) & C{< for all | # C [since | # Ps(|) & C].

Now consider decision problem D=[0, x, y], where x|=&1 if| # A, x|==1 if | # B, and x|=0 if | � A _ B; and where y|=&=2 if| # A, y|==1 if | # B, y|=&1 if | # C, and y|=0 if | � A _ B _ C.By continuity, it is possible to choose =1>0 sufficiently small and =2>0sufficiently smaller than =1 such that

C|, s[D]=[0] for all | # A [by (i) and (ii)]

C|, t[D]/[x, y] for all | # A [by (iii) and (iv)]

C|, s[D]/[x, y] for all | # B [by (v) and (vi)]

C|, s[D]/[0, x] for all | # C [by (vii)].

Now z # Cs*[D] implies z|==1 for all | # B and z|=0 for all | � B; butv # Ct*[D] implies v|�&=2 for all | # A, z|�=1 for all | # B, and z|�0for all | � A _ B; thus zo* v, contradicting valuable information. K

5. Conclusion

The central idea of this paper is that it must be possible to make aconnection between preferences and the logical properties of belief andknowledge which underlie standard representations of information ineconomics. This connection is of interest whether one believes that a deci-sion maker's beliefs and knowledge exist prior to his preferences, or if oneis a pure subjectivist, who believes that all statements about belief andknowledge must be reducible to observed choices and thus preferences.With the exception of the independent work of Lipman [15] (discussedbelow) this paper is the first in the economics literature to pursue thisapproach.

This general approach must be distinguished from the particular direc-tion which has been pursued. This paper follows a literature in economicswhich has examined the economic consequences of weakening partitioninformation (see footnote 2). Modica and Rustichini [17] have argued thatthe forms of non-partitional information structures studied fail to capturethe idea of ``unawareness'' or unforeseen contingencies which was at leastpart of their motivation. Lipman [15] examines the consequences ofweakening the more basic assumption��implicit in the distributivityassumption (B1) in this paper��that decision makers believe all the logical



consequences of everything they believe. But whatever system of belief andknowledge is chosen for study, this paper has illustrated a generalapproach to relating that system to assumptions about preferences.

In defense of the particular approach of this paper, the results relatedstandard weakenings of the assumption of partition information to naturalproperties of preferences. Because decision makers do not understand thestate space, coherence, and valuable information cannot be thought of asnormative rules for the decision maker. But from the viewpoint of an exter-nal observer, they test whether a decision maker's choices could berationalized by an appropriate meta-ordering.

An additional rationale for studying the properties of preferencesconsidered here was provided by Geanakoplos [10]. Many results in infor-mation economics rely only on the fact that ex post decisions do not makethe decision maker worse off according to some appropriate ex antecriterion, i.e., in the language of this paper, they are coherent. For example,``no trade'' theorems show that if there are no ex ante gains from trade thenoptimal ex post decisions, following the arrival of information, will not leadto trade. Coherence, and thus weaker non-partitional information struc-tures, are sufficient for many economic arguments. On the other hand, theresults of this paper show that in dynamic settings, nothing significantlyweaker than information partitions will suffice for standard informationtheoretic results.

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5. A. Brandenburger, E. Dekel, and J. Geanakoplos, Correlated equilibrium withgeneralized information structures, Games Econ. Behavior 4 (1992), 182�201.

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10. J. Geanakoplos, ``Game Theory without Partitions,'' Cowles Foundation DiscussionPaper *914, Yale University, 1989.

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